Question

An XYZ-3000 is a front-end processor to five mainframe computers at RST U. There are 64 incoming phone lines to the XYZ-3000. In how many ways can the front-end processor assign lines to computers so that 8 are directed to \(C_{1}, 14\) to \(C_{2}, 17\) to \(C_{3}, 16\) to \(C_{4},\) and the remaining to \(C_{5} ?\) A program called TUNE monitors the performance of a computer system. Suppose each user is assigned to one of the 64 memory areas when first logged onto the system. TUNE samples memory areas or partitions when a user is first logged on to the system to decide how to assign the new user memory. How many can this be done if TUNE samples 17 of 64 system partitions? How many if the one fixed partition \(Z\) is always excluded from the sample? How many if two fixed partitions are alwavs chosen?

Short answer

1. Use multinomial coefficient: \( \frac{64!}{8! \cdot 14! \cdot 17! \cdot 16! \cdot 9!} \). 2. 64 choose 17: \( \binom{64}{17} \). 3. 63 choose 17: \( \binom{63}{17} \). 4. 62 choose 15: \( \binom{62}{15} \).

Step-by-step solution

Assigning Lines to Computers

The problem is to find the number of ways to assign 64 phone lines to five computers with specific numbers of lines assigned to each. The partitions for each computer are: 8 lines to \( C_1 \), 14 lines to \( C_2 \), 17 lines to \( C_3 \), 16 lines to \( C_4 \), and the remaining 9 lines to \( C_5 \). This can be calculated using multinomial coefficients, represented by \( \binom{64}{8,14,17,16,9} \).The formula for multinomial coefficients is:\[\binom{n}{k_1, k_2, k_3, \ldots, k_m} = \frac{n!}{k_1! k_2! k_3! \ldots k_m!}\]where \( n = 64 \) and \( k_1 = 8 \), \( k_2 = 14 \), \( k_3 = 17 \), \( k_4 = 16 \), \( k_5 = 9 \).

Calculate the Multinomial Coefficient

Plug the values into the multinomial formula:\[\binom{64}{8,14,17,16,9} = \frac{64!}{8! \cdot 14! \cdot 17! \cdot 16! \cdot 9!}\]This gives us the number of ways to distribute the phone lines to the computers.

Sample Memory Partitions with No Exclusions

The next part of the question asks about sampling 17 out of 64 memory areas, with no exclusions. This is a simple combination problem:\[\binom{64}{17}\]Calculate this value to find the number of ways to sample 17 partitions.

Sample Memory Partitions with One Exclusion

If one fixed partition \( Z \) is always excluded, we effectively have 63 partitions to choose from. We need to sample 17 out of these 63:\[\binom{63}{17}\]This calculates the number of ways to choose 17 partitions from 63.

Sample Memory Partitions with Two Chosen

If two specific partitions are always chosen, we must choose 15 more from the remaining 62 partitions, since we've already chosen 2:\[\binom{62}{15}\]This gives the number of ways to complete the sample of 17 partitions including the fixed two.

Key concepts

Multinomial Coefficient

Understanding multinomial coefficients is crucial in scenarios where you need to allocate or arrange items into different groups. Imagine you have a certain number of items, and these need to be split into several subgroups, each with a specific number of items. This is precisely what a multinomial coefficient helps you compute.

In mathematical terms, for arranging a total of \(n\) items into \(m\) groups with specific sizes, the multinomial coefficient is represented as \(\binom{n}{k_1, k_2, \ldots, k_m}\). Here, \(k_1, k_2, \ldots, k_m\) are the sizes of the groups, and they sum up to \(n\).

The formula for calculating a multinomial coefficient is:
\[ \binom{n}{k_1, k_2, \ldots, k_m} = \frac{n!}{k_1! k_2! \cdots k_m!} \]
This equation uses factorials (notated by \(!\)), which multiply a series of descending natural numbers. For example, the factorial of 5, \(5!\), is equal to \(5 \times 4 \times 3 \times 2 \times 1 = 120\).

• Use this concept when dividing a set into non-identical parts.
• Helpful in scenarios like distributing tasks, resources, or, in this case, phone lines among computers.

This method brilliantly handles complicated arrangements making tasks technically feasible.

Combination Formula

The combination formula is a fundamental tool in combinatorics. It is used to determine the number of ways to choose a subset of items from a larger set without regard to the order of selection. This is especially useful when you only need certain amounts of a larger group, such as memory partitions where only a few are sampled from the total.

The general formula for a combination is given by:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, \(n\) represents the total number of items, and \(r\) denotes the number of items to choose.

• \(\binom{64}{17}\) computes the selection of 17 partitions from 64 systems.
• Subsequently, \(\binom{63}{17}\) signifies picking 17 when one specific partition is excluded.
• For always including two partitions, choose 15 more from the remaining 62 with \(\binom{62}{15}\).

The power of this formula lies in its ability to ignore permutations and focus on the combination subset, making scenarios like memory selection manageable.

Memory Partition Sampling

Sampling within a memory context involves selecting a portion of available memory spaces to be utilized or tested, such as allocating memory areas for active user connections. Memory partition sampling can ensure optimal performance or resource allocation by testing particular areas before actual usage.

For the TUNE scenario, think of it this way:

• Without any exclusions, you're sampling a few (17 out of 64) to understand usage patterns or performance.
• If the slot \(Z\) is reserved or never used, adapt the sample size to a reduced number \(63\) while still picking the same amount \(17\).
• Alternatively, setting two partitions always active means assessing the remaining slots (15 from 62) as you've reserved space for the constant pair.

This process is a smart way to simulate potential loads or evaluate system effectiveness in changing memory use scenarios.

Ultimately, this kind of sampling assists in smartly managing and tuning systems, offering efficiency and ensuring stable operations in a dynamically changing environment.